Asymptotic enumeration and limit laws for graphs of fixed genus
Résumé
It is shown that the number of labelled graphs with $n$ vertices that can be embedded in the orientable surface $\mathbb{S}_g$ of genus $g$ grows asymptotically like $ c^{(g)}n^{5(g-1)/2-1}\gamma^n n! $, where $c^{(g)} >0$, and $\gamma \approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case $g=0$, obtained by Giménez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in $\mathbb{S}_g$ has a unique 2-connected component of linear size with high probability.